On uniqueness of elastic scattering from a cavity

In this paper, we consider a time-harmonic plane elastic wave incident on a cavity which can be regarded as a local perturbation below the plane with wide-ranging applications as engine inlet ducts, cracks, gaps and so on.

For electromagnetic incident waves, there are considerable research results in literature. For instance, Ammari, Bao and Wood proved uniqueness and existence of electromagnetic cavity problems respectively to TE and TM polarization by variational approaches in [1], and by integral equation methods in [2]. They also extend similar results for Maxwell’s equations in [3] by variational approaches. Recently, Bao et al established a stability result explicitly dependent on wave number for TE polarization based on Fourier analysis in energy space in [4] and TM polarization in [5]. Numerical methods for large cavities can be seen in [6, 7, 8].

Compared to electromagnetic scattering, elastic scattering also has wide applications in seismology and geophysics such as [9, 10, 11] and attracts more attention recently. However, elastic scattering problems have not been studied intensively due to their inherent difficulties arising from the governing equations and boundary conditions. For periodic surface, Arens studied quasi-periodic Green tensor and elastic potentials in [12] and used Rayleigh expansions and Rellich identities to prove uniqueness and integral equation methods to prove existence in [13]. Elschner and Hu applied variational approaches to get similar results in [14]. For general unbounded rough surfaces, Arens studied Green tensor, elastic potentials and upward radiation condition (UPRC), and proved uniqueness by integral equation methods and existence by integral equation methods, see [15, 16, 17]. Elschner and Hu [18] gave an equivalent form of UPRC and proved uniqueness and existence by variational approaches. They [19] also considered solvability in weighted Sobolev spaces and proved existence and uniqueness of elastic scattering from unbounded rough surface with an incident wave. Moreover, Hu, Yuan and Zhao [20] combined variation formula and integral equation to prove uniqueness and existence of two-dimensions local perturbed surface scattering problem with Lipschitz graph. For cavities, Hu, Li and Zhao considered elastic scattering in three-dimensions and proved the uniqueness in [21] for Dirichlet boundary conditions.

Inverse cavity problem is to reconstruct the shape of a cavity by measured far-field data on the artificial boundary. In general, to reconstruct a shape, Fr$\acute{\rm e}$chet derivative is commonly used. For instance, Bao, Gao and Li considered TE and TM polarization in [22] and proved uniqueness for lossy medium and studied the domain derivative and local stability for inverse electromagnetic cavity. A similar results for Maxwell’s equations was extended in [23]. Bao and Lai [24] proposed an optimization scheme to design a cavity for radar cross section reduction. Recently, Hu, Yuan, Zhao considered domain derivative for the inverse elastic scattering by a locally perturbed rough surface in [20]. To the authors’ best knowledge, research on inverse elastic scattering from cavities is still at its infant stage.

This paper intends to extend electromagnetic scattering from a cavity to elastic scattering. By employing transparent boundary conditions given by Dirichlet to Neumann (DtN) and Neumann to Dirichlet (NtD) operators, the boundary value problems can be reduced into a bounded domain so that the Fredholm alternative can be applied. Then, the variational approach is used to prove existence and uniqueness of the elastic scattering problem in two-dimensions with Dirichlet condition or Neumann condition. For Dirichlet problem, the existence in two-dimensions is actually trivial compared to three dimensions in [21]. However, we can obtain the uniqueness for two-dimensions while it is still open for three dimensions. Neumann problem is similar to Dirichlet problem with additional difficulties for the NtD operator. On inverse elastic cavity problem, this paper applies the method of changing variable to give the Fr$\acute{\rm e}$chet derivative for Dirichlet problem and Neumann problem, respectively. The difficulty for inverse Neumann problem lies in the boundary value problem reduced by it in Section 2 has inhomogeneous Neumann boundary condition by directly applying NtD operator. Hence, in order to obtain the TBC deduced by the DtN operator, we use an artificial boundary to rewrite the boundary value problem for Neumann case.

The paper is outlined as follows. In Section 2, the direct problems are reduced into variation problems in a bounded domain by DtN operator and NtD operator. In Section 3 and Section 4, existence and uniqueness for Dirichlet problem and Neumann problem are examined respectively. Moreover, the solutions to variation problems can be extended to satisfy Kurpradze-Sommerfeld radiation condition and Navier equations in upper half-space. In section 5, the Fr$\acute{\rm e}$chet derivatives for Dirichlet problem and Neumann problem are given respectively and a local stability result is given for Dirichlet problem. Conclusion is given in Section 6.

Consider a time-harmonic plane wave incident on a cavity, which is shown in Figure 1, denoted by $D$ with boundary $\Gamma\cup S$, where $\Gamma\subset\{x_{2}=0\}$ is bounded and $S$ is assumed Lipschitz continuous. It is noted that $S$ is not necessary a graph of some function. In addition, denote $\Gamma^{c}=\{x_{2}=0\}\backslash\Gamma$. The medium is assumed to be homogeneous.

Suppose the incident time-harmonic plane wave takes the form

where

Define the compressional and shear wave numbers by

For simplicity, throughout this paper, it assumes $c_{p}=k_{p}$, $c_{s}=0$. Consider total wave field $u$ which satisfies the following Navier equations with Lam$\acute{\rm e}$ constants $\lambda>0$, $\mu>0$ and frequency $\omega>0$

For convenience, denote by $\Delta^{*}u=\mu\Delta u+(\mu+\lambda)\nabla(\nabla\cdot u)$. This paper considers two kinds of boundary conditions, i.e., Dirichlet condition

and Neumann condition

where the differential operator is defined $Tu:=\mu\partial_{n}u+(\lambda+\mu)\vec{n}{\rm div}\,u$. Consider the Helmholtz decomposition for $u$, i.e.,

with

where $\overrightarrow{\text{curl}}\,u=(\partial_{2}u_{1},-\partial_{1}u_{2})^{\top}$ and $\text{curl}\,u=\partial_{1}u_{2}-\partial_{2}u_{1}$. Then, $\phi$ and $\psi$ satisfy the homogeneous Helmholtz equations

The total wave field consists of incident wave $u_{i}$, reflected wave $u_{r}$ and scattering wave $u_{s}$, i.e., $u=u_{i}+u_{r}+u_{s}$. Here, the reflected wave $u_{r}$ satisfies

It is easy to know that the reflected wave exists and can be expressed by the following functions

with $\alpha=k_{p}\sin{\theta},\,\beta=k_{p}\cos{\theta}$ and $\eta=\sqrt{k_{s}^{2}-\alpha^{2}}$ which are linearly independent solutions to Navier equations. Combing the Dirichlet boundary condition gives

And the Neumann boundary condition (2.8) yields

For the scattering wave $u_{s}$, it is required to satisfy the following half-plane Kupradze-Sommerfeld radiation condition,

and

with $r=|x|$ for $\phi=-\frac{i}{k^{2}_{p}}{\rm div}u_{s}$, $\psi=\frac{i}{k^{2}_{s}}{\rm curl}u_{s}$.

Now we can state the following two problems respectively.

Dirichlet problem: Find $u$ which satisfies (2.1)-(2.2), where $u_{s}$ satisfies Kupradze-Sommerfeld radiation condition (2.9)-(2.10).

Neumann problem: Find $u$ which satisfies (2.1) and (2.3), where $u_{s}$ satisfies Kupradze-Sommerfeld radiation condition (2.9)-(2.10).

To investigate the above two problems in an unbounded domain, it is necessary to introduce transparent boundary conditions (TBC) to reduce the model problems into a bounded domain.

By Helmholtz decomposition, we have $u_{s}$ satisfying (2.4) with corresponding $\phi$, $\psi$ satisfying (2.5)-(2.6). Applying the Fourier transform to (2.6) with respect to $x_{1}$ and using radiation condition gives

with

Denote by $\hat{u}(\xi)=\mathcal{F}u(\xi)$ the Fourier transformation of $u$ with respect to $x_{1}$. Then the Fourier transformation of $u$ is given by

Let $x_{2}=0$,

which implies

Inserting (2.13) into (2.11) arrives at the following representation for $u_{s}$:

in $\{x_{2}>0\}$ with

Recalling the definition of differential operator $T$, we have

Combing (2.14)-(2.15) and direct calculation implies that

where

Then the Dirichlet to Neumann operator can be defined by

So $Tu_{s}=\mathcal{T}u_{s}$ on $\Gamma$. Noting that $u_{i}+u_{r}=0$ on $\{x_{2}=0\}$ and $u=0$ on $\Gamma^{c}$, then TBC for Dirichlet problem can be given by

By TBC, Dirichlet problem can be reformulated as

Denote $V=H^{1}_{S}(D)^{2}=\{u\in H^{1}(D)^{2}:u=0\,\mathrm{on}\,S\}$. Then for $u,\,v\in V$, the Betti formula gives

where

Define a sesquilinear form $B\,:V\times V\to\mathbb{C}$ as

Now we obtain variation formula of Dirichlet boundary value problem.

Variation problem 1: find $u\in V$ such that $B(u,v)=(g,v)_{\Gamma},\,\forall v\in V$.

It is similar to give TBC and variation formula corresponding to Neumann problem. Note that in this case, it is not $u_{s}$ but $Tu_{s}$, which has compact support on $\Gamma$. So $\hat{u}_{s}$ may not be integrable on $\{x_{2}=0\}$. Thus, (2.13) and (2.16) may not hold in this case. However, $u_{s}$ can be represented by $\widehat{Tu_{s}}(\xi,0)$ similarly. According to (2.14), we obtain

Combing (2.11) and (2.19) yields

Take $x_{2}=0$, then $P,\,S$ can be represented by $\widehat{Tu_{s}}(\xi,0)$. Insert the representation in (2.11), we have

Taking inverse Fourier transformation and $x_{2}=0$ gives

Hence NtD operator $\varLambda$ can be defined by

which implies

Denote $h=u_{r}+u_{i}$ and $g=-(\Delta^{*}+\omega^{2})h$. Then the Neumann problem can be reformulated as

Define the function space

where

and $\tilde{u}$ is zero extension of $u$. In fact, it is clear that $H^{-1/2}_{0}(\Gamma)^{2}$ is the dual space of $H^{1/2}(\Gamma)^{2}$ in [17]. The sesquilinear form $\mathcal{B}$: $W\times W\rightarrow\mathbb{C}$ is defined by

Now we can give the following variation formula by the Betti formula.

Variation problem 2: find $u_{s}\in W$ such that

Next two sections consider existence and uniqueness of solutions to the above two variation problems. Moreover, they can be extended to $\{x_{2}>0\}$ satisfying Kurpradze-Sommerfeld radiation condition.

Since this problem is in a bounded domain with Lipschitz continuous boundary, we have the well-known compact embedding result that $H^{1}(D)$ is a compact subset of $L^{2}(D)$. Hence uniqueness of variation problem and Fredholm alternative yield existence directly. To this end, we show that the continuity of DtN operator results in the continuity of the sesquilinear form $B$, which is stated in the following lemma with the proof omitted.

In order to use Fredholm alternative, it needs to show the sesquilinear form satisfies the G$\mathring{\rm a}$rding inequality. Take real part of $B$,

Considering (2.18) gives

It turns out that

By Plancherel identity,

where $\Re\,M=(M+\bar{M}^{\top})/2$. However, the sign of $-\Re\,\left\langle\mathcal{T}u,u\right\rangle_{\Gamma}$ is undeterminied because it lacks the result that $-\Re\,M(\xi)$ is positive definite or negative definite. Fortunately, for sufficiently large $|\xi|$, we know $-\Re\,M(\xi)$ is positive definite, given in Lemma 3.2.

The proof is also omitted here as it is trivial compared as the results in [18]. Noting that

and using Lemma 3.2, we have

Therefore, it suffices to estimate the intergal in $\{|\xi|\leq k_{s}\}$. The following inequality is similar as the three-dimensions case in [20].

Next the uniqueness and existence of the variation problem can be derived by Lemma 3.3 and the Fredholm alternative.

Remark. Theorem 3.1 shows that there admits a unique solution $u$ to Variation problem 1 in bounded domain $D$. We can extend $u_{s}=u-u_{i}-u_{r}$ to $\{x_{2}>0\}$ by (2.14), though it may not satisfy Kupradze-Sommerfeld radiation condition. In fact, the condition that $u_{s}$ satisfies (2.14) is weaker than Kupradze-Sommerfeld radiation condition of half-plane (2.9)-(2.10) (see [15]). But we can extend $u_{s}$ by

where $G_{D}(x,y)$ is the half-space Green tensor for Dirichlet problem. According to the property of Green tensor we can verify that the corresponding $u$ is the solution to Drichlet problem in $D\cup\{x_{2}\geq 0\}$ (see [15]). Then by representation theorem we know that any solution to Dirichlet problem in $D\cup\{x_{2}\geq 0\}$ satisfies (3.3), which means the extension is unique.

Recalling the two variation problems in Section 2, we can find that the main difference between Dirichlet problem and Neumann one is that $M$ in (2.17) is replaced by $M^{-1}$, i.e.,

where

This section shows that $M^{-1}$ has similar properties to $M$ though it is more complicated. Before proceed to the uniqueness, we give the following lemma to show that $\varLambda$ is continuous, which implies the sesquilinear form is also continuous.

Then we consider $-\Re\,M^{-1}$ similarly as Section 2.

Using the above lemma, we can estimate $\Re\,\mathcal{B}(u,u)$ and then get the following inequality.